标题： Sobolev空间和均匀边界表示 显示英文标题

标题： Sobolev spaces and uniform boundary representations

摘要： 我们证明了在适当的分数Sobolev空间$W^{s,p}$上，某些边界表示的统一有界性，其中$p>1$适用于任意Gromov双曲群。 这些是$L^p$的闭子空间，并且在$p=2$的情况下尤其为Hilbert空间。 这种构造允许我们，通过适当选择$p$，用统一有界的表示来逼近平凡表示。 当所考虑的双曲群具有Property (T)时，这种现象在等距表示的设置中没有对应情况。 关键是引入了一个在赋予共形结构（按照Mineyev的方式）的度量空间上的度量共形算子的概念，以及由Cayley变换诱导的Sobolev空间同构的度量类似物。 显示英文摘要

摘要： We prove uniform boundedness of certain boundary representations on appropriate fractional Sobolev spaces $W^{s,p}$ with $p>1$ for arbitrary Gromov hyperbolic groups. These are closed subspaces of $L^p$ and in particular Hilbert spaces in the case $p=2$. This construction allows us, for an appropriate choice of $p$, to approximate the trivial representation through uniformly bounded representations. This phenomenon does not have analogue in the setting of isometric representations whenever the hyperbolic group considered has the Property (T). The key is the introduction of a notion of metrically conformal operator on a metric space endowed with a conformal structure \`{a} la Mineyev and a metric analogue of the isomorphisms of Sobolev spaces induced by the Cayley transform.